Verified Answer :
- Set the function to zero and divide by the lead coefficient: \(x^2 - \frac{4}{5}x = -\frac{1}{5}\).
- Complete the square by adding \(\left(-\frac{2}{5}\right)^2 = \frac{4}{25}\) to both sides.
- Simplify to \(\left(x - \frac{2}{5}\right)^2 = -\frac{1}{25}\).
- Solve for \(x\) to find the roots: \(x = \boxed{\frac{2}{5} \pm \frac{1}{5}i}\).
Explanation
- Analysis of the Quadratic Function
We are tasked with finding the zeros of the quadratic function \(f(x) = -5x^2 + 4x - 1\) by completing the square. A zero of a function occurs where the function value is equal to zero, so we set the equation \(f(x) = 0\), which gives us:
\[ -5x^2 + 4x - 1 = 0 \]
This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = -5\), \(b = 4\), and \(c = -1\). Our goal is to manipulate this equation into the form \((x + h)^2 = k\) to easily solve for \(x\).
- Simplifying and Isolating the Constant Term
To complete the square, we first ensure the coefficient of the \(x^2\) term is 1. We achieve this by dividing the entire equation by \(-5\):
\[ \frac{-5x^2}{-5} + \frac{4x}{-5} - \frac{1}{-5} = \frac{0}{-5} \]
\[ x^2 - \frac{4}{5}x + \frac{1}{5} = 0 \]
Next, we isolate the constant term by moving it to the right side of the equation:
\[ x^2 - \frac{4}{5}x = -\frac{1}{5} \]
- Completing the Square
To complete the square, we need to add a term to the left side that makes it a perfect square trinomial. This term is the square of half the coefficient of the \(x\) term. The coefficient of \(x\) is \(-\frac{4}{5}\). Half of this is:
\[ \frac{1}{2} \cdot \left(-\frac{4}{5}\right) = -\frac{2}{5} \]
Squaring this result, we get:
\[ \left(-\frac{2}{5}\right)^2 = \frac{4}{25} \]
We add \(\frac{4}{25}\) to both sides of the equation:
\[ x^2 - \frac{4}{5}x + \frac{4}{25} = -\frac{1}{5} + \frac{4}{25} \]
Now, the left side is a perfect square:
\[ \left(x - \frac{2}{5}\right)^2 = -\frac{5}{25} + \frac{4}{25} = -\frac{1}{25} \]
- Solving for the Zeros
Now we solve for \(x\) by taking the square root of both sides. Since we are taking the square root of a negative number, we will obtain complex solutions involving the imaginary unit \(i = \sqrt{-1}\):
\[ x - \frac{2}{5} = \pm \sqrt{-\frac{1}{25}} \]
\[ x - \frac{2}{5} = \pm \frac{\sqrt{-1}}{\sqrt{25}} \]
\[ x - \frac{2}{5} = \pm \frac{1}{5}i \]
Finally, we add \(\frac{2}{5}\) to both sides to isolate \(x\):
\[ x = \frac{2}{5} \pm \frac{1}{5}i \]
Thus, the two zeros of the function are \(x = \frac{2}{5} + \frac{1}{5}i\) and \(x = \frac{2}{5} - \frac{1}{5}i\).
Examples
Completing the square is a powerful algebraic technique used far beyond basic algebra, appearing frequently in physics and engineering. For instance, when analyzing the motion of an object under specific forces (like a simple harmonic oscillator), the equations of motion are often quadratic. Finding the equilibrium point or the roots of such equations allows us to predict the system's stability and oscillation frequency. By mastering this method, you gain a versatile tool for solving complex equations and analyzing systems in real-world scenarios.